nLab infinitary tensor product



Measure and probability theory

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monoidal categories

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In higher category theory

Limits and colimits



In a category with all products, one can take finite products of objects, and hence one has a monoidal category. One can however also take infinite products. In some situations, such as in categorical probability, one is interesting in taking a infinitary version of a tensor product.

In measure-theoretic probability theory, such a construction is provided by Kolmogorov's extension theorem, and the same idea can be used in general:

  • In a category with products, an infinite product can be expressed as a cofiltered limit of finite products.
  • Similarly, one can define an infinitary tensor product as a cofiltered limit, if it exists, of finite tensor products.


Let (C,,I)(C,\otimes,I) be a symmetric monoidal category equipped with a distinguished map del X:XIdel_X:X\to I for each object II. (This happens for example if CC is cartesian semicartesian, or if it has a Markov or copy-discard structure.)

Let JJ be a possibly infinite set, and let (X i) iJ(X_i)_{i\in J} be a JJ-indexed collection of objects of CC.

For every finite subset FJF\subseteq J, denote by

iFX i \bigotimes_{i\in F} X_i

the tensor product of the family (X i) iF(X_i)_{i\in F}. (Note that FF is not ordered, but the ordering of the terms in the tensor product does not matter up to coherent isomorphism, since CC is symmetric monoidal.)

Consider now finite subsets FGF\subseteq G of JJ. We have a canonical “projection” morphism defined as follows. First of all, for iGi\in G, let

Y i={X i iF; I iF. Y_i = \begin{cases} X_i & i\in F ; \\ I & i\notin F. \end{cases}

Similarly, let e i:X iY ie_i:X_i\to Y_i be

e i={id X i iF; del X i iF. e_i = \begin{cases} id_{X_i} & i\in F ; \\ del_{X_i} & i\notin F. \end{cases}

The map π G,F\pi_{G,F} is given by the tensor product of the e ie_i as follows, where the isomorphism on the right is given by the unitors of the monoidal category.

Let now Fin(J)Fin(J) be the poset of finite subsets of GG. The union of finite subsets is finite, so Fin(J)Fin(J) is a directed set, hence a filtered category. The functor Fin(J) opCFin(J)^op\to C mapping the inclusion FGF\subseteq G to π G,F\pi_{G,F} defined above is hence a cofiltered diagram.

We say that an object is an infintary tensor product of the family (X i) iJ(X_i)_{i\in J} in CC, and denote it by X JX_J, if

  • It is a cofiltered limit of the functor Fin(J) opCFin(J)^op\to C described above, and moreover
  • For every object AA, the functor A:CCA\otimes-:C\to C preserves this limit.


  • In a cartesian monoidal category, every infinite product, if it exists, is an infinitary tensor product.

  • In a Markov category, Kolmogorov products are particular infinitary products compatible with the copy-discard structure. In particular, the category BorelStoch has all countable infinitary tensor products.

  • Reversing all arrows, the infinite tensor product of rings is defined as the filtered colimit of all the finitary tensor products. (The canonical arrows IXI\to X are the units of the rings.)

See also


Last revised on January 30, 2024 at 20:39:55. See the history of this page for a list of all contributions to it.